So you want to solve, for each $k$, the following equation:
$$(n+1)x = k + c ~~ \forall x \in \left[\frac{k}{n}, \frac{k+1}{n}\right)$$
The solution is:
$$x^* = \frac{k+c}{n+1}$$
This solution is feaseble if $x^*$ is in the set $\left[\frac{k}{n}, \frac{k+1}{n}\right)$. Then: